Samrat Goswami

Tight cocone: a water-tight surface reconstructor

Surface reconstruction from unorganized sample points is an important problem in computer graphics, computer aided design, medical imaging and solid modeling. Recently a few algorithms have been developed that have theoretical guarantee of computing a topologically correct and geometrically close surface under certain condition on sampling density. Unfortunately, this sampling condition is not always met in practice due to noise, non-smoothness or simply due to inadequate sampling. This leads to undesired holes and other artifacts in the output surface. Certain CAD applications such as creating a prototype from a model boundary require a water tight surface, i.e., no hole should be allowed in the surface. In this paper we describe a simple algorithm called Tight Cocone that works on an initial mesh generated by a popular surface reconstruction algorithm and fills up all holes to output a water-tight surface. In doing so, it does not introduce any extra points and produces a triangulated surface interpolating the input sample points. In support of our method we present experimental results with a number of difficult data sets.

Shape segmentation and matching with flow discretization

Geometric shapes are identified with their features. For computational purposes a concrete mathematical definition of features is required. In this paper we use a topological approach, namely dynamical systems, to define features of shapes. To exploit this definition algorithmically we assume that a point sample of the shape is given as input from which features of the shape have to be approximated. We translate our definition of features to the discrete domain while mimicking the set-up developed for the continuous shapes. Experimental results show that our algorithms segment shapes in two and three dimensions into so-called features quite effectively. Further, we develop a shape matching algorithm that takes advantage of our robust feature segmentation step.

Provable surface reconstruction from noisy samples

We present an algorithm for surface reconstruction in presence of noise. We show that, under a reasonable noise model, the algorithm has theoretical guarantees. Actual performance of the algorithm is illustrated by our experimental results.

Shape dimension and approximation from samples

Abstract. There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi-based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to. Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory.

Identifying flat and tubular regions of a shape by unstable manifolds

We present an algorithm to identify the flat and tubular regions of a three dimensional shape from its point sample. We consider the distance function to the input point cloud and the Morse structure induced by it on R3. Specifically we focus on the index 1 and index 2 saddle points and their unstable manifolds. The unstable manifolds of index 2 saddles are one dimensional whereas those of index 1 saddles are two dimensional. Mapping these unstable manifolds back onto the surface, we get the tubular and flat regions. The computations are carried out on the Voronoi diagram of the input points by approximating the unstable manifolds with Voronoi faces. We demonstrate the performance of our algorithm on several point sampled objects.

Topology Based Selection and Curation of Level Sets

The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent quantitative processing and visualization. For an initial selection of an isosurface, guided by contour tree data structures, we detect the topological features by computing stable and unstable manifolds of the critical points of the distance function induced by the isosurface. We further enhance the description of these features by associating geometric attributes with them. We then rank the attributed features and provide a handle to them for curation of the topological anomalies.

Shape segmentation and matching from noisy point clouds

We present the implementation results of a shape segmentation technique and an associated shape matching method whose input is a point sample from the shape. The sample is allowed to be noisy in the sense that they may scatter around the boundary of the shape instead of lying exactly on it. The algorithm is simple and mostly combinatorial in that it builds a single data structure, the Delaunay triangulation of the point set, and groups the tetrahedra to form the segments. A small set of weighted points are derived from the segments which are used as signatures to match shapes. Experimental results establish the effectiveness of the method in practice.

Undersampling and oversampling in sample based shape modeling

Shape modeling is an integral part of many visualization problems. Recent advances in scanning technology and a number of surface reconstruction algorithms have opened up a new paradigm for modeling shapes from samples. Many of the problems currently faced in this modeling paradigm can be traced back to two anomalies in sampling, namely undersampling and oversampling. Boundaries, non-smoothness and small features create undersampling problems, whereas oversampling leads to too many triangles. We use Voronoi cell geometry as a unified guide to detect undersampling and oversampling. We apply these detections in surface reconstruction and model simplification. Guarantees of the algorithms can be proved. The authors show the success of the algorithms empirically on a number of interesting data sets.

Multi-component heart reconstruction from volumetric imaging

Computer Tomography (CT) and in particular super fast, 64 and 256 detector CT has rapidly advanced over recent years, such that high resolution cardiac imaging has become a reality. In this paper, we briefly introduce a framework that we have built to construct three dimensional (3D) finite-element and boundary element mesh models of the human heart directly from high resolution CT imaging data. Although, the overall IMAGING-MODELING framework consists of image processing, geometry processing and meshing algorithms, our main focus in this paper will revolve around three key geometry processing steps which are parts of the so-called IMAGING-MODELING framework. These three steps are geometry cleanup or CURATION, anatomy guided annotation or SEGMENTATION and construction of GENERALIZED OFFSET SURFACE. These three algorithms, due to the very nature of the computation involved, can also be thought as parts of a more generalized modeling technique, namely geometric modeling with distance function. As part of the results presented in the paper, we will show that our algorithms are robust enough to effectively deal with the challenges posed by the real-world patient CT data collected from our radiologist collaborators.

Efficient Delaunay Mesh Generation from Sampled Scalar Functions

Many modern research areas face the challenge of meshing level sets of sampled scalar functions. While many algorithms focus on ensuring geometric qualities of the output mesh, recent attention has been paid to building topologically accurate Delaunay conforming meshes of any level set from such volumetric data.